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Graphical Inequalities for the Linear Ordering Polytope Jean-Paul Doignon Universit Libre de Bruxelles Joint work with Samuel Fiorini and Gwenal Joret Universit Libre de Bruxelles 1 Graphical Inequalities for the Linear Ordering

  1. Rephrasing the Main Problem The linear ordering polytope P Z LO has the vertices x L , for L ∈ Π ; find the facets of the linear ordering polytope P Z LO . And the usual comment: the problem is hopeless! A manageable solution would give P = NP . 10

  2. Rephrasing the Main Problem The linear ordering polytope P Z LO has the vertices x L , for L ∈ Π ; find the facets of the linear ordering polytope P Z LO . And the usual comment: the problem is hopeless! A manageable solution would give P = NP . 10

  3. Origins of the Problem In mathematical psychology/economics: Guilbaud (1953), Block and Marschak (1960). In discrete mathematics: Megiddo (1977). In operations research: Grötschel, Jünger and Reinelt (1985). In voting theory: Saari (1999). 11

  4. Examples of Facet-defining Inequalities for P n LO Remember our obvious necessary conditions. Theorem The following affine (linear) inequalities on R Z ⋉ Z define facets: p ij ≥ 0 (trivial inequalities) , p ij + p jk + p ki ≤ 2 (triangular inequalities) . A first scheme of nonobvious facets is due independently to Cohen and Falmagne (1978, published in 1990), Grötschel, Jünger and Reinelt (1985). 12

  5. Examples of Facet-defining Inequalities for P n LO Remember our obvious necessary conditions. Theorem The following affine (linear) inequalities on R Z ⋉ Z define facets: p ij ≥ 0 (trivial inequalities) , p ij + p jk + p ki ≤ 2 (triangular inequalities) . A first scheme of nonobvious facets is due independently to Cohen and Falmagne (1978, published in 1990), Grötschel, Jünger and Reinelt (1985). 12

  6. Examples of Facet-defining Inequalities for P n LO Remember our obvious necessary conditions. Theorem The following affine (linear) inequalities on R Z ⋉ Z define facets: p ij ≥ 0 (trivial inequalities) , p ij + p jk + p ki ≤ 2 (triangular inequalities) . A first scheme of nonobvious facets is due independently to Cohen and Falmagne (1978, published in 1990), Grötschel, Jünger and Reinelt (1985). 12

  7. First Example of Fence Inequality The following inequality is facet-defining: x as + x bt + x cu − ( x at + x bs ) − ( x au + x cs ) − ( x bu + x ct ) ≤ 1 . s t u a b c 13

  8. The Fence Inequality In general, let X , Y ⊂ Z with X ∩ Y = ∅ , | X | = | Y | , f : X → Y a bijective mapping (we keep the notation throughout). Y f X 14

  9. The Fence Inequality Definition The fence inequality is � � � � x i f ( i ) − x i f ( j ) + x j f ( i ) ≤ 1 . i ∈ X i , j ∈ X , i � = j Theorem (Cohen and Falmagne, 1978; Grötschel, Jünger and Reinelt, 1985) For | X | ≥ 3 , the fence inequality defines a facet of the linear ordering polytope P n LO . 15

  10. The Fence Inequality Definition The fence inequality is � � � � x i f ( i ) − x i f ( j ) + x j f ( i ) ≤ 1 . i ∈ X i , j ∈ X , i � = j Theorem (Cohen and Falmagne, 1978; Grötschel, Jünger and Reinelt, 1985) For | X | ≥ 3 , the fence inequality defines a facet of the linear ordering polytope P n LO . 15

  11. A Structural Generalization of the Fence Inequality Several steps: McLennan (1990), Fishburn (1990), Koppen (1991), etc. leading to a marvelous result by Koppen (1995). Let G = ( V , E ) be a (simple) graph. The stability number α ( G ) of G is the largest number of vertices no two of which are adjacent. Assume f : X → Y as before, and moreover V = X . Definition The graphical inequality of G reads � � − ( x i , f ( j ) + x j , f ( i ) ) ≤ α ( G ) . x i , f ( i ) i ∈ V { i , j }∈ E 16

  12. A Structural Generalization of the Fence Inequality Several steps: McLennan (1990), Fishburn (1990), Koppen (1991), etc. leading to a marvelous result by Koppen (1995). Let G = ( V , E ) be a (simple) graph. The stability number α ( G ) of G is the largest number of vertices no two of which are adjacent. Assume f : X → Y as before, and moreover V = X . Definition The graphical inequality of G reads � � − ( x i , f ( j ) + x j , f ( i ) ) ≤ α ( G ) . x i , f ( i ) i ∈ V { i , j }∈ E 16

  13. A Structural Generalization of the Fence Inequality Several steps: McLennan (1990), Fishburn (1990), Koppen (1991), etc. leading to a marvelous result by Koppen (1995). Let G = ( V , E ) be a (simple) graph. The stability number α ( G ) of G is the largest number of vertices no two of which are adjacent. Assume f : X → Y as before, and moreover V = X . Definition The graphical inequality of G reads � � − ( x i , f ( j ) + x j , f ( i ) ) ≤ α ( G ) . x i , f ( i ) i ∈ V { i , j }∈ E 16

  14. A Structural Generalization of the Fence Inequality Several steps: McLennan (1990), Fishburn (1990), Koppen (1991), etc. leading to a marvelous result by Koppen (1995). Let G = ( V , E ) be a (simple) graph. The stability number α ( G ) of G is the largest number of vertices no two of which are adjacent. Assume f : X → Y as before, and moreover V = X . Definition The graphical inequality of G reads � � − ( x i , f ( j ) + x j , f ( i ) ) ≤ α ( G ) . x i , f ( i ) i ∈ V { i , j }∈ E 16

  15. A Structural Generalization of the Fence Inequality Several steps: McLennan (1990), Fishburn (1990), Koppen (1991), etc. leading to a marvelous result by Koppen (1995). Let G = ( V , E ) be a (simple) graph. The stability number α ( G ) of G is the largest number of vertices no two of which are adjacent. Assume f : X → Y as before, and moreover V = X . Definition The graphical inequality of G reads � � − ( x i , f ( j ) + x j , f ( i ) ) ≤ α ( G ) . x i , f ( i ) i ∈ V { i , j }∈ E 16

  16. An Example of Graphical Inequality Example For the graph a b d c with the bijection f : a �→ s , b �→ t , c �→ u , d �→ v , we get the inequality x as + x bt + x cu + x dv − ( x at + x bs ) − ( x bu + x ct ) − ( x cv + x du ) − ( x ds + x av ) ≤ 2 . 17

  17. An Example of Graphical Inequality Example For the graph a b d c with the bijection f : a �→ s , b �→ t , c �→ u , d �→ v , we get the inequality x as + x bt + x cu + x dv − ( x at + x bs ) − ( x bu + x ct ) − ( x cv + x du ) − ( x ds + x av ) ≤ 2 . 17

  18. An Example of Graphical Inequality Example For the graph a b d c with the bijection f : a �→ s , b �→ t , c �→ u , d �→ v , we get the inequality x as + x bt + x cu + x dv − ( x at + x bs ) − ( x bu + x ct ) − ( x cv + x du ) − ( x ds + x av ) ≤ 2 . 17

  19. Main Result in Koppen (1995) Theorem (Koppen, 1995) The graphical inequality of G is valid for the linear ordering polytope. It defines a facet if and only if G is different from K 2 , connected, and stability critical. Definition A graph is stability critical when its stability number increases whenever any of its edges is deleted. 18

  20. Main Result in Koppen (1995) Theorem (Koppen, 1995) The graphical inequality of G is valid for the linear ordering polytope. It defines a facet if and only if G is different from K 2 , connected, and stability critical. Definition A graph is stability critical when its stability number increases whenever any of its edges is deleted. 18

  21. Main Result in Koppen (1995) Theorem (Koppen, 1995) The graphical inequality of G is valid for the linear ordering polytope. It defines a facet if and only if G is different from K 2 , connected, and stability critical. Definition A graph is stability critical when its stability number increases whenever any of its edges is deleted. 18

  22. An Example of Stability-Critical Graph Examples Delete any edge: Thus: the 5-cycle is stability critical but the 6-cycle is not. 19

  23. An Example of Stability-Critical Graph Examples Delete any edge: Thus: the 5-cycle is stability critical but the 6-cycle is not. 19

  24. A Weighted Generalization of the Fence Inequality Independently: Leung and Lee (1994), Suck (1992). Theorem For | X | ≥ 3 , the reinforced fence inequality t ( t + 1 ) � � t x i , f ( i ) − ( x i , f ( j ) + x j , f ( i ) ) ≤ 2 i ∈ X i , j ∈ X , i � = j defines a facet of P n LO if and only if the constant value t satisfies 1 ≤ t ≤ | X | − 2 . 20

  25. Our Contribution (D., F. and J.) Schematically: fence inequality graphical inequality of reinforced fence inequality a stability critical graph (of a complete graph) A common generalization? 21

  26. Our Contribution (D., F. and J.) Schematically: fence inequality graphical inequality of reinforced fence inequality a stability critical graph (of a complete graph) A common generalization? 21

  27. Our Contribution (D., F. and J.) Schematically: fence inequality graphical inequality of reinforced fence inequality a stability critical graph (of a complete graph) A common generalization? 21

  28. Our Contribution (D., F. and J.) Schematically: fence inequality graphical inequality of reinforced fence inequality a stability critical graph (of a complete graph) A common generalization? 21

  29. Preparing a General Graphical Inequality Let ( G , µ ) be a weighted graph, with G = ( V , E ) and µ : V → Z . Definition For S ⊆ V , the worth (or net weight) w ( S ) equals the total weight µ ( S ) minus the number of edges in S . A subset of S is tight if it maximizes the worth. Notation α ( G , µ ) = max S ⊆ V w ( S ) . Remark If µ = 1 (constant weight 1 ), then α ( G , 1 ) = α ( G ) . Thus α ( G , µ ) is a true generalization of α ( G ) . 22

  30. Preparing a General Graphical Inequality Let ( G , µ ) be a weighted graph, with G = ( V , E ) and µ : V → Z . Definition For S ⊆ V , the worth (or net weight) w ( S ) equals the total weight µ ( S ) minus the number of edges in S . A subset of S is tight if it maximizes the worth. Notation α ( G , µ ) = max S ⊆ V w ( S ) . Remark If µ = 1 (constant weight 1 ), then α ( G , 1 ) = α ( G ) . Thus α ( G , µ ) is a true generalization of α ( G ) . 22

  31. Preparing a General Graphical Inequality Let ( G , µ ) be a weighted graph, with G = ( V , E ) and µ : V → Z . Definition For S ⊆ V , the worth (or net weight) w ( S ) equals the total weight µ ( S ) minus the number of edges in S . A subset of S is tight if it maximizes the worth. Notation α ( G , µ ) = max S ⊆ V w ( S ) . Remark If µ = 1 (constant weight 1 ), then α ( G , 1 ) = α ( G ) . Thus α ( G , µ ) is a true generalization of α ( G ) . 22

  32. Preparing a General Graphical Inequality Let ( G , µ ) be a weighted graph, with G = ( V , E ) and µ : V → Z . Definition For S ⊆ V , the worth (or net weight) w ( S ) equals the total weight µ ( S ) minus the number of edges in S . A subset of S is tight if it maximizes the worth. Notation α ( G , µ ) = max S ⊆ V w ( S ) . Remark If µ = 1 (constant weight 1 ), then α ( G , 1 ) = α ( G ) . Thus α ( G , µ ) is a true generalization of α ( G ) . 22

  33. Preparing a General Graphical Inequality Let ( G , µ ) be a weighted graph, with G = ( V , E ) and µ : V → Z . Definition For S ⊆ V , the worth (or net weight) w ( S ) equals the total weight µ ( S ) minus the number of edges in S . A subset of S is tight if it maximizes the worth. Notation α ( G , µ ) = max S ⊆ V w ( S ) . Remark If µ = 1 (constant weight 1 ), then α ( G , 1 ) = α ( G ) . Thus α ( G , µ ) is a true generalization of α ( G ) . 22

  34. Preparing a General Graphical Inequality Let ( G , µ ) be a weighted graph, with G = ( V , E ) and µ : V → Z . Definition For S ⊆ V , the worth (or net weight) w ( S ) equals the total weight µ ( S ) minus the number of edges in S . A subset of S is tight if it maximizes the worth. Notation α ( G , µ ) = max S ⊆ V w ( S ) . Remark If µ = 1 (constant weight 1 ), then α ( G , 1 ) = α ( G ) . Thus α ( G , µ ) is a true generalization of α ( G ) . 22

  35. Examples of Tight Sets Example For the pentagon with µ = 1 , here are tight sets: Remember that tight sets S maximize w ( S ) = µ ( S ) − || S || . 23

  36. Graphical Inequalities Let ( G , µ ) be a weighted graph, with G = ( V , E ) and µ : V → Z . Definition Let f : X → Y be bijective with X , Y ⊂ Z , X ∩ Y = ∅ , and assume V = X . The graphical inequality of ( G , µ ) reads � � µ ( i ) x i , f ( i ) − ( x i , f ( j ) + x j , f ( i ) ) ≤ α ( G , µ ) . i ∈ V { i , j }∈ E Proposition The graphical inequality is always valid for the linear ordering polytope P Z LO . 24

  37. Graphical Inequalities Let ( G , µ ) be a weighted graph, with G = ( V , E ) and µ : V → Z . Definition Let f : X → Y be bijective with X , Y ⊂ Z , X ∩ Y = ∅ , and assume V = X . The graphical inequality of ( G , µ ) reads � � µ ( i ) x i , f ( i ) − ( x i , f ( j ) + x j , f ( i ) ) ≤ α ( G , µ ) . i ∈ V { i , j }∈ E Proposition The graphical inequality is always valid for the linear ordering polytope P Z LO . 24

  38. Graphical Inequalities Let ( G , µ ) be a weighted graph, with G = ( V , E ) and µ : V → Z . Definition Let f : X → Y be bijective with X , Y ⊂ Z , X ∩ Y = ∅ , and assume V = X . The graphical inequality of ( G , µ ) reads � � µ ( i ) x i , f ( i ) − ( x i , f ( j ) + x j , f ( i ) ) ≤ α ( G , µ ) . i ∈ V { i , j }∈ E Proposition The graphical inequality is always valid for the linear ordering polytope P Z LO . 24

  39. Graphical Inequalities Let ( G , µ ) be a weighted graph, with G = ( V , E ) and µ : V → Z . Definition Let f : X → Y be bijective with X , Y ⊂ Z , X ∩ Y = ∅ , and assume V = X . The graphical inequality of ( G , µ ) reads � � µ ( i ) x i , f ( i ) − ( x i , f ( j ) + x j , f ( i ) ) ≤ α ( G , µ ) . i ∈ V { i , j }∈ E Proposition The graphical inequality is always valid for the linear ordering polytope P Z LO . 24

  40. An Example of Graphical Inequality Example Consider X = { a , b , c , d } , Y = { s , t , u , v } , and the bijection f : a �→ s , b �→ t , c �→ u , d �→ v . Take the graph a b 2 1 5 2 d c Its graphical inequality is 2 x as + x bt + 2 x cu + 5 x dv − ( x at + x bs ) − ( x au + x cs ) − ( x av + x ds ) − ( x bu + x ct ) − ( x cv + x du ) ≤ 6 . 25

  41. An Example of Graphical Inequality Example Consider X = { a , b , c , d } , Y = { s , t , u , v } , and the bijection f : a �→ s , b �→ t , c �→ u , d �→ v . Take the graph a b 2 1 5 2 d c Its graphical inequality is 2 x as + x bt + 2 x cu + 5 x dv − ( x at + x bs ) − ( x au + x cs ) − ( x av + x ds ) − ( x bu + x ct ) − ( x cv + x du ) ≤ 6 . 25

  42. An Example of Graphical Inequality Example Consider X = { a , b , c , d } , Y = { s , t , u , v } , and the bijection f : a �→ s , b �→ t , c �→ u , d �→ v . Take the graph a b 2 1 5 2 d c Its graphical inequality is 2 x as + x bt + 2 x cu + 5 x dv − ( x at + x bs ) − ( x au + x cs ) − ( x av + x ds ) − ( x bu + x ct ) − ( x cv + x du ) ≤ 6 . 25

  43. Facet-defining Graphs Definition A weighted graph is facet defining or a FDG if its graphical inequality defines a facet of P n LO . Examples 1 1 1 2 2 1 1 2 2 1 1 2 1 1 1 2 1 2 2 1 1 2 3 1 2 3 3 1 1 2 3 3 1 2 2 1 26

  44. Facet-defining Graphs Definition A weighted graph is facet defining or a FDG if its graphical inequality defines a facet of P n LO . Examples 1 1 1 2 2 1 1 2 2 1 1 2 1 1 1 2 1 2 2 1 1 2 3 1 2 3 3 1 1 2 3 3 1 2 2 1 26

  45. A Subsidiary Problem Problem To understand FDGs, e.g. to classify them. Remark FDGs include connected, stability critical graphs with more than 2 vertices. Hard (although only partial) results were obtained in classifying the latter graphs, see e.g. Lovász (1993).   Remark   Another weighted, generalization of stability critical graphs   is investigated by Lipták and Lovász (2000, 2001). 27

  46. A Subsidiary Problem Problem To understand FDGs, e.g. to classify them. Remark FDGs include connected, stability critical graphs with more than 2 vertices. Hard (although only partial) results were obtained in classifying the latter graphs, see e.g. Lovász (1993).   Remark   Another weighted, generalization of stability critical graphs   is investigated by Lipták and Lovász (2000, 2001). 27

  47. A Subsidiary Problem Problem To understand FDGs, e.g. to classify them. Remark FDGs include connected, stability critical graphs with more than 2 vertices. Hard (although only partial) results were obtained in classifying the latter graphs, see e.g. Lovász (1993).   Remark   Another weighted, generalization of stability critical graphs   is investigated by Lipták and Lovász (2000, 2001). 27

  48. A Subsidiary Problem Problem To understand FDGs, e.g. to classify them. Remark FDGs include connected, stability critical graphs with more than 2 vertices. Hard (although only partial) results were obtained in classifying the latter graphs, see e.g. Lovász (1993).   Remark   Another weighted, generalization of stability critical graphs   is investigated by Lipták and Lovász (2000, 2001). 27

  49. A Subsidiary Problem Problem To understand FDGs, e.g. to classify them. Remark FDGs include connected, stability critical graphs with more than 2 vertices. Hard (although only partial) results were obtained in classifying the latter graphs, see e.g. Lovász (1993).   Remark   Another weighted, generalization of stability critical graphs   is investigated by Lipták and Lovász (2000, 2001). 27

  50. An Unsatisfactory Answer Theorem Let ( G , µ ) be a weighted graph with more than two vertices. Then ( G , µ ) is a FDG if and only if for each nonzero valuation λ : V ( G ) ∪ E ( G ) → Z there is a tight set T of ( G , µ ) with � � λ ( t ) + λ ( e ) � = 0 . v ∈ T e ∈ E ( T ) Remark We lack a simple characterization of FDGs. 28

  51. An Unsatisfactory Answer Theorem Let ( G , µ ) be a weighted graph with more than two vertices. Then ( G , µ ) is a FDG if and only if for each nonzero valuation λ : V ( G ) ∪ E ( G ) → Z there is a tight set T of ( G , µ ) with � � λ ( t ) + λ ( e ) � = 0 . v ∈ T e ∈ E ( T ) Remark We lack a simple characterization of FDGs. 28

  52. An Unsatisfactory Answer Theorem Let ( G , µ ) be a weighted graph with more than two vertices. Then ( G , µ ) is a FDG if and only if for each nonzero valuation λ : V ( G ) ∪ E ( G ) → Z there is a tight set T of ( G , µ ) with � � λ ( t ) + λ ( e ) � = 0 . v ∈ T e ∈ E ( T ) Remark We lack a simple characterization of FDGs. 28

  53. Sketch of the proof Take f : X → Y as in the definition of the graphical inequality. Take the restrictions to X × Y of all linear orderings L of Z . The resulting relations from X to Y coincide with the “biorders” from X to Y (Doignon, Ducamp and Falmagne, 1984). is defined in R X × Y (Christophe, The biorder polytope P X × Y Bio Doignon and Fiorini, 2004). The restriction L �→ L | X × Y induces a “polytope projection” LO → P X × Y P n . Bio Etc. 29

  54. Sketch of the proof Take f : X → Y as in the definition of the graphical inequality. Take the restrictions to X × Y of all linear orderings L of Z . The resulting relations from X to Y coincide with the “biorders” from X to Y (Doignon, Ducamp and Falmagne, 1984). is defined in R X × Y (Christophe, The biorder polytope P X × Y Bio Doignon and Fiorini, 2004). The restriction L �→ L | X × Y induces a “polytope projection” LO → P X × Y P n . Bio Etc. 29

  55. Sketch of the proof Take f : X → Y as in the definition of the graphical inequality. Take the restrictions to X × Y of all linear orderings L of Z . The resulting relations from X to Y coincide with the “biorders” from X to Y (Doignon, Ducamp and Falmagne, 1984). is defined in R X × Y (Christophe, The biorder polytope P X × Y Bio Doignon and Fiorini, 2004). The restriction L �→ L | X × Y induces a “polytope projection” LO → P X × Y P n . Bio Etc. 29

  56. Sketch of the proof Take f : X → Y as in the definition of the graphical inequality. Take the restrictions to X × Y of all linear orderings L of Z . The resulting relations from X to Y coincide with the “biorders” from X to Y (Doignon, Ducamp and Falmagne, 1984). is defined in R X × Y (Christophe, The biorder polytope P X × Y Bio Doignon and Fiorini, 2004). The restriction L �→ L | X × Y induces a “polytope projection” LO → P X × Y P n . Bio Etc. 29

  57. Sketch of the proof Take f : X → Y as in the definition of the graphical inequality. Take the restrictions to X × Y of all linear orderings L of Z . The resulting relations from X to Y coincide with the “biorders” from X to Y (Doignon, Ducamp and Falmagne, 1984). is defined in R X × Y (Christophe, The biorder polytope P X × Y Bio Doignon and Fiorini, 2004). The restriction L �→ L | X × Y induces a “polytope projection” LO → P X × Y P n . Bio Etc. 29

  58. Sketch of the proof Take f : X → Y as in the definition of the graphical inequality. Take the restrictions to X × Y of all linear orderings L of Z . The resulting relations from X to Y coincide with the “biorders” from X to Y (Doignon, Ducamp and Falmagne, 1984). is defined in R X × Y (Christophe, The biorder polytope P X × Y Bio Doignon and Fiorini, 2004). The restriction L �→ L | X × Y induces a “polytope projection” LO → P X × Y P n . Bio Etc. 29

  59. First Results on Facet Defining Graphs Theorem For any FDG ( G , µ ) , the graph G is 2 -connected. Theorem If ( G , µ ) is a FDG, so is ( G , deg − µ ) . [Here ( deg − µ )( v ) = deg ( v ) − µ ( v ) .] Thus most stability critical graphs produce two FDGs: one with µ = 1 , another one with µ = deg − 1 . Let’s go back to stability critical graphs (FDGs when µ = 1 ). 30

  60. First Results on Facet Defining Graphs Theorem For any FDG ( G , µ ) , the graph G is 2 -connected. Theorem If ( G , µ ) is a FDG, so is ( G , deg − µ ) . [Here ( deg − µ )( v ) = deg ( v ) − µ ( v ) .] Thus most stability critical graphs produce two FDGs: one with µ = 1 , another one with µ = deg − 1 . Let’s go back to stability critical graphs (FDGs when µ = 1 ). 30

  61. First Results on Facet Defining Graphs Theorem For any FDG ( G , µ ) , the graph G is 2 -connected. Theorem If ( G , µ ) is a FDG, so is ( G , deg − µ ) . [Here ( deg − µ )( v ) = deg ( v ) − µ ( v ) .] Thus most stability critical graphs produce two FDGs: one with µ = 1 , another one with µ = deg − 1 . Let’s go back to stability critical graphs (FDGs when µ = 1 ). 30

  62. First Results on Facet Defining Graphs Theorem For any FDG ( G , µ ) , the graph G is 2 -connected. Theorem If ( G , µ ) is a FDG, so is ( G , deg − µ ) . [Here ( deg − µ )( v ) = deg ( v ) − µ ( v ) .] Thus most stability critical graphs produce two FDGs: one with µ = 1 , another one with µ = deg − 1 . Let’s go back to stability critical graphs (FDGs when µ = 1 ). 30

  63. The Defect of Stability Critical Graphs For any graph G = ( V , E ) (no weight here), define its defect δ ( G ) = | V | − 2 α ( G ) . Consider here a connected, stability critical graph G . Theorem (Erdös and Gallai, 1961) δ ( G ) ≥ 0 . Theorem (Hajnal, 1965) Any vertex v of G satisfies deg ( v ) ≤ δ ( G ) + 1 . Corollary (Hajnal, 1965) δ ( G ) = 0 ⇐ ⇒ G = K 2 ; δ ( G ) = 1 ⇐ ⇒ G is an odd cycle. Odd-cycles coincide with odd subdivisions of K 3 . 31

  64. The Defect of Stability Critical Graphs For any graph G = ( V , E ) (no weight here), define its defect δ ( G ) = | V | − 2 α ( G ) . Consider here a connected, stability critical graph G . Theorem (Erdös and Gallai, 1961) δ ( G ) ≥ 0 . Theorem (Hajnal, 1965) Any vertex v of G satisfies deg ( v ) ≤ δ ( G ) + 1 . Corollary (Hajnal, 1965) δ ( G ) = 0 ⇐ ⇒ G = K 2 ; δ ( G ) = 1 ⇐ ⇒ G is an odd cycle. Odd-cycles coincide with odd subdivisions of K 3 . 31

  65. The Defect of Stability Critical Graphs For any graph G = ( V , E ) (no weight here), define its defect δ ( G ) = | V | − 2 α ( G ) . Consider here a connected, stability critical graph G . Theorem (Erdös and Gallai, 1961) δ ( G ) ≥ 0 . Theorem (Hajnal, 1965) Any vertex v of G satisfies deg ( v ) ≤ δ ( G ) + 1 . Corollary (Hajnal, 1965) δ ( G ) = 0 ⇐ ⇒ G = K 2 ; δ ( G ) = 1 ⇐ ⇒ G is an odd cycle. Odd-cycles coincide with odd subdivisions of K 3 . 31

  66. The Defect of Stability Critical Graphs For any graph G = ( V , E ) (no weight here), define its defect δ ( G ) = | V | − 2 α ( G ) . Consider here a connected, stability critical graph G . Theorem (Erdös and Gallai, 1961) δ ( G ) ≥ 0 . Theorem (Hajnal, 1965) Any vertex v of G satisfies deg ( v ) ≤ δ ( G ) + 1 . Corollary (Hajnal, 1965) δ ( G ) = 0 ⇐ ⇒ G = K 2 ; δ ( G ) = 1 ⇐ ⇒ G is an odd cycle. Odd-cycles coincide with odd subdivisions of K 3 . 31

  67. The Defect of Stability Critical Graphs For any graph G = ( V , E ) (no weight here), define its defect δ ( G ) = | V | − 2 α ( G ) . Consider here a connected, stability critical graph G . Theorem (Erdös and Gallai, 1961) δ ( G ) ≥ 0 . Theorem (Hajnal, 1965) Any vertex v of G satisfies deg ( v ) ≤ δ ( G ) + 1 . Corollary (Hajnal, 1965) δ ( G ) = 0 ⇐ ⇒ G = K 2 ; δ ( G ) = 1 ⇐ ⇒ G is an odd cycle. Odd-cycles coincide with odd subdivisions of K 3 . 31

  68. The Defect of Stability Critical Graphs For any graph G = ( V , E ) (no weight here), define its defect δ ( G ) = | V | − 2 α ( G ) . Consider here a connected, stability critical graph G . Theorem (Erdös and Gallai, 1961) δ ( G ) ≥ 0 . Theorem (Hajnal, 1965) Any vertex v of G satisfies deg ( v ) ≤ δ ( G ) + 1 . Corollary (Hajnal, 1965) δ ( G ) = 0 ⇐ ⇒ G = K 2 ; δ ( G ) = 1 ⇐ ⇒ G is an odd cycle. Odd-cycles coincide with odd subdivisions of K 3 . 31

  69. Odd Subdivisions of a Graph Definition An odd subdivision of a graph G is obtained by replacing a number of edges of G with odd paths (of various lengthes). Example (an 11-cycle) Theorem (Andrásfai, 1967) The connected stability critical graphs with defect 2 are the odd-subdivision of K 4 . 32

  70. Odd Subdivisions of a Graph Definition An odd subdivision of a graph G is obtained by replacing a number of edges of G with odd paths (of various lengthes). Example (an 11-cycle) Theorem (Andrásfai, 1967) The connected stability critical graphs with defect 2 are the odd-subdivision of K 4 . 32

  71. Odd Subdivisions of a Graph Definition An odd subdivision of a graph G is obtained by replacing a number of edges of G with odd paths (of various lengthes). Example (an 11-cycle) Theorem (Andrásfai, 1967) The connected stability critical graphs with defect 2 are the odd-subdivision of K 4 . 32

  72. The Basis Theorem for Stability Critical Graphs Theorem (Lovász, 1978) For any natural number δ > 0 , there is a finite collection S δ of graphs such that G is a connected stability critical graph with δ ( G ) = δ ⇐ ⇒ G is an odd-subdivision of some graph in S δ . Examples S 1 : S 2 : 33

  73. The Basis of Stability Critical Graphs with defect 3 Among the graphs in S 3 , we show only those with minimum degree 3: There are 7 other graphs in S δ (according to Gwen). 34

  74. The Basis of Stability Critical Graphs with defect 3 Among the graphs in S 3 , we show only those with minimum degree 3: There are 7 other graphs in S δ (according to Gwen). 34

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